The second main statement is that this invariant polynomial serves to provide a differential (Lie integration) construction of [c]ℝ[c]_{\mathbb{R}}:

for any choice of connection∇\nabla on a GG-principal bundle P→XP \to X we have the curvature 2-form F∇∈Ω2(P,𝔤)F_\nabla \in \Omega^2(P, \mathfrak{g}) and fed into the invariant polynomial this yields an nn-form

E. Cartan conjectured that there should be a general result implying that the homology of the classical Lie groups is the same as the homology of a product of odd-dimensional spheres. In particular, he lists the Poincare polynomial?s for classical simple compact Lie groups.

at the end (1951) of an era of deRham cohomology dominence (prior to Serre’s thesis) abstracted the differential geometric approach of Chern-Weil and the Weil algebraW(𝔤)W(\mathfrak{g}) to the dg-algebra context with his notion of 𝔤\mathfrak{g}-algebras AA . This involves what is known sometimes as the Cartan calculus. In addition to the differential dd of differential forms on a principal bundle, Cartan abstracts the inner product aka contraction of differential forms with vector fields XX and the Lie derivativeℒX\mathcal{L}_X with respect to vector fields. that is, he posits 3 operators on a differential-graded-commutative alggebra (dgca):

dd of degree 1, iXi_X of degree -1 and LXL_X of degree 0 for X in 𝔤\mathfrak{g} subject to the relations:

[ιX,ιY]=ι[X,Y][\iota_X,\iota_Y] = \iota_{[X,Y]}

[ℒX,ιY]=ι[X,Y] [\mathcal{L}_X,\iota_Y]= \iota_{[X,Y]}

and perhaps most useful

ℒX=dιX+iιXd\mathcal{L}_X = d \iota_X + i\iota_X d

This is what he terms a 𝔤\mathfrak{g}-algebra.

For Cartan, an infinitesimal connection on a principal bundleP→XP \to X are projectors (at each point pp of PP) ϕp:TpP→Tpvert\phi_p: T_p P\to T_p^{vert} equivariant with repect to the GG-action. This can be abstracted to a morphism

In general, this will not respect the differentials, hence not be a morphism of dg-algebras. In fact, the deviation gives the curvature of the connection: the curvature tensor is the map h↦ddRA(h)−A(dCEh)h\mapsto d_{dR} A(h)-A(d_{CE} h).

Jim: HAVE TO BREAK OFF NOW - WHAT WILL COME NEXT IS the Weil algebraW(𝔤)W(\mathfrak{g}) as a Cartan 𝔤\mathfrak{g}-algebra